Standard quotients of homogeneous spaces: the case of plane waves

It is well known, for instance, from the Calabi-Markus phenomenon, that a homogeneous space $X=G/H$ of a Lie group $G$ does not always admit a compact manifold modeled on it (i.e., possessing a $(G,X)$-structure). When the structure is complete, such a manifold is identified with a compact quotient of the homogeneous space. A compact quotient of a homogeneous space is said to be standard if the action of the fundamental group $\Gamma$ extends to a simple and transitive action of a connected Lie subgroup $L$ of $G$ (containing $\Gamma$ as a lattice). This is the case for certain flat affine geometries. Indeed, a classical result by Goldman, Fried, and Kamishima shows that a compact quotient of Minkowski space (which is identified with the homogeneous space $O(1,n) \ltimes \mathbb{R}^{n+1} / O(1,n)$) is standard, generalizing the Riemannian Bieberbach theorem to the Lorentzian signature. It turns out that looking for standard quotients is an easier problem when studying the existence of compact quotients of homogeneous spaces. I will discuss the case of certain homogeneous Lorentzian manifolds, called plane waves, which are deformations and generalizations of Minkowski space.  I will conclude with some interesting consequences and questions. This is a joint work with M. Hanounah, I. Kath, and A. Zeghib.